Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $n = \dfrac{-3z - 27}{9z - 27} \times \dfrac{-6z + 18}{z^2 + 8z - 9} $
Answer: First factor the quadratic. $n = \dfrac{-3z - 27}{9z - 27} \times \dfrac{-6z + 18}{(z + 9)(z - 1)} $ Then factor out any other terms. $n = \dfrac{-3(z + 9)}{9(z - 3)} \times \dfrac{-6(z - 3)}{(z + 9)(z - 1)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac{ -3(z + 9) \times -6(z - 3) } { 9(z - 3) \times (z + 9)(z - 1) } $ $n = \dfrac{ 18(z + 9)(z - 3)}{ 9(z - 3)(z + 9)(z - 1)} $ Notice that $(z - 3)$ and $(z + 9)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac{ 18\cancel{(z + 9)}(z - 3)}{ 9(z - 3)\cancel{(z + 9)}(z - 1)} $ We are dividing by $z + 9$ , so $z + 9 \neq 0$ Therefore, $z \neq -9$ $n = \dfrac{ 18\cancel{(z + 9)}\cancel{(z - 3)}}{ 9\cancel{(z - 3)}\cancel{(z + 9)}(z - 1)} $ We are dividing by $z - 3$ , so $z - 3 \neq 0$ Therefore, $z \neq 3$ $n = \dfrac{18}{9(z - 1)} $ $n = \dfrac{2}{z - 1} ; \space z \neq -9 ; \space z \neq 3 $